$f_n$ is a sequence of real functions defined on $[0,1]$.
For all n, we have
i) $ f_n(0)=f'_n(0)=0$
ii) $f''$ is continuous, and $|f''(x)|\leq 1$ for all $x \in [0,1]$
Prove that ${f_n}$ has a subsequece converging uniformly on $[0,1]$
My Solution:
I have proved that $f_n$ is pointwise bounded.
Thus, for any $x\in [0,1]$, we can find a sequence ${x_k}$ converging to x and a subsequence of $f_n$ (denote this subsequence as $\{g_n\}$) converging for every point in $x_k$. i.e. given any $x_k$, $ \{ g_n(x_k) \}$ converges as $n\rightarrow \infty$
I want to prove that $ \{g_n\}$ converges uniformly on $[0,1]$
I am considering the following:
$|g_n(x)-g_m(x)|\leq|g_n(x)-g_n(x_k)|+|g_n(x_k)-g_m(x_k)|+|g_m(x_k)-g_m(x)|$
I think the first part and the third part con be controlled below $\epsilon$ by continuous and convergence of $x_k$ and the second part can be controlled by pointwise convergence. But when writing the proof, I found that the selection of $m$ and $n$ depends on $k$ , and that the selection of $k$ depends on $m$ and $n$.
Did I ignore anything? Or can I just say, "for any $\epsilon$ we can find $x_k$, $N_k$ such that $m, n> N_k$ implies that all conditions hold"?